Simplify ((10r+6)/(6r^2+42r))÷((15r^2-21r-18)/(3r-6))

$\frac{10 r + 6}{6 r^{2} + 42 r} \div \frac{15 r^{2} - 21 r - 18}{3 r - 6}$

To divide by a fraction, multiply by its reciprocal.

$\frac{10 r + 6}{6 r^{2} + 42 r} \cdot \frac{3 r - 6}{15 r^{2} - 21 r - 18}$

Factor $2$ out of $10 r + 6$ .

Factor $2$ out of $10 r$ .

$\frac{2 (5 r) + 6}{6 r^{2} + 42 r} \cdot \frac{3 r - 6}{15 r^{2} - 21 r - 18}$

Factor $2$ out of $6$ .

$\frac{2 (5 r) + 2 (3)}{6 r^{2} + 42 r} \cdot \frac{3 r - 6}{15 r^{2} - 21 r - 18}$

Factor $2$ out of $2 (5 r) + 2 (3)$ .

$\frac{2 (5 r + 3)}{6 r^{2} + 42 r} \cdot \frac{3 r - 6}{15 r^{2} - 21 r - 18}$

Factor $6 r$ out of $6 r^{2} + 42 r$ .

Factor $6 r$ out of $6 r^{2}$ .

$\frac{2 (5 r + 3)}{6 r (r) + 42 r} \cdot \frac{3 r - 6}{15 r^{2} - 21 r - 18}$

Factor $6 r$ out of $42 r$ .

$\frac{2 (5 r + 3)}{6 r (r) + 6 r (7)} \cdot \frac{3 r - 6}{15 r^{2} - 21 r - 18}$

Factor $6 r$ out of $6 r (r) + 6 r (7)$ .

$\frac{2 (5 r + 3)}{6 r (r + 7)} \cdot \frac{3 r - 6}{15 r^{2} - 21 r - 18}$

Cancel the common factors.

Factor $2$ out of $6 r (r + 7)$ .

$\frac{2 (5 r + 3)}{2 (3 r (r + 7))} \cdot \frac{3 r - 6}{15 r^{2} - 21 r - 18}$

Cancel the common factor.

$\frac{2 (5 r + 3)}{2 (3 r (r + 7))} \cdot \frac{3 r - 6}{15 r^{2} - 21 r - 18}$

Rewrite the expression.

$\frac{5 r + 3}{3 r (r + 7)} \cdot \frac{3 r - 6}{15 r^{2} - 21 r - 18}$

Cancel the common factor of $3 r - 6$ and $15 r^{2} - 21 r - 18$ .

Factor $3$ out of $3 r$ .

$\frac{5 r + 3}{3 r (r + 7)} \cdot \frac{3 (r) - 6}{15 r^{2} - 21 r - 18}$

Factor $3$ out of $- 6$ .

$\frac{5 r + 3}{3 r (r + 7)} \cdot \frac{3 (r) + 3 (- 2)}{15 r^{2} - 21 r - 18}$

Factor $3$ out of $3 (r) + 3 (- 2)$ .

$\frac{5 r + 3}{3 r (r + 7)} \cdot \frac{3 (r - 2)}{15 r^{2} - 21 r - 18}$

Cancel the common factors.

Factor $3$ out of $15 r^{2}$ .

$\frac{5 r + 3}{3 r (r + 7)} \cdot \frac{3 (r - 2)}{3 (5 r^{2}) - 21 r - 18}$

Factor $3$ out of $- 21 r$ .

$\frac{5 r + 3}{3 r (r + 7)} \cdot \frac{3 (r - 2)}{3 (5 r^{2}) + 3 (- 7 r) - 18}$

Factor $3$ out of $3 (5 r^{2}) + 3 (- 7 r)$ .

$\frac{5 r + 3}{3 r (r + 7)} \cdot \frac{3 (r - 2)}{3 (5 r^{2} - 7 r) - 18}$

Factor $3$ out of $- 18$ .

$\frac{5 r + 3}{3 r (r + 7)} \cdot \frac{3 (r - 2)}{3 (5 r^{2} - 7 r) + 3 (- 6)}$

Factor $3$ out of $3 (5 r^{2} - 7 r) + 3 (- 6)$ .

$\frac{5 r + 3}{3 r (r + 7)} \cdot \frac{3 (r - 2)}{3 (5 r^{2} - 7 r - 6)}$

Cancel the common factor.

$\frac{5 r + 3}{3 r (r + 7)} \cdot \frac{3 (r - 2)}{3 (5 r^{2} - 7 r - 6)}$

Rewrite the expression.

$\frac{5 r + 3}{3 r (r + 7)} \cdot \frac{r - 2}{5 r^{2} - 7 r - 6}$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot - 6 = - 30$ and whose sum is $b = - 7$ .

Factor $- 7$ out of $- 7 r$ .

$\frac{5 r + 3}{3 r (r + 7)} \cdot \frac{r - 2}{5 r^{2} - 7 (r) - 6}$

Rewrite $- 7$ as $3$ plus $- 10$

$\frac{5 r + 3}{3 r (r + 7)} \cdot \frac{r - 2}{5 r^{2} + (3 - 10) r - 6}$

Apply the distributive property.

$\frac{5 r + 3}{3 r (r + 7)} \cdot \frac{r - 2}{5 r^{2} + 3 r - 10 r - 6}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{5 r + 3}{3 r (r + 7)} \cdot \frac{r - 2}{(5 r^{2} + 3 r) - 10 r - 6}$

Factor out the greatest common factor (GCF) from each group.

$\frac{5 r + 3}{3 r (r + 7)} \cdot \frac{r - 2}{r (5 r + 3) - 2 (5 r + 3)}$

Factor the polynomial by factoring out the greatest common factor, $5 r + 3$ .

$\frac{5 r + 3}{3 r (r + 7)} \cdot \frac{r - 2}{(5 r + 3) (r - 2)}$

Cancel the common factor of $5 r + 3$ .

Cancel the common factor.

$\frac{5 r + 3}{3 r (r + 7)} \cdot \frac{r - 2}{(5 r + 3) (r - 2)}$

Rewrite the expression.

$\frac{1}{3 r (r + 7)} \cdot \frac{r - 2}{r - 2}$

Multiply $\frac{1}{3 r (r + 7)}$ by $\frac{r - 2}{r - 2}$ .

$\frac{r - 2}{3 r (r + 7) (r - 2)}$

Cancel the common factor of $r - 2$ .

Cancel the common factor.

$\frac{r - 2}{3 r (r + 7) (r - 2)}$

Rewrite the expression.

$\frac{1}{3 r (r + 7)}$

Simplify ((10r+6)/(6r^2+42r))÷((15r^2-21r-18)/(3r-6))

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